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Properties

Label 201.4.m.a
Level $201$
Weight $4$
Character orbit 201.m
Analytic conductor $11.859$
Analytic rank $0$
Dimension $320$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,Mod(4,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.m (of order \(33\), degree \(20\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(16\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 2 q^{2} - 96 q^{3} + 66 q^{4} - 4 q^{5} + 27 q^{6} + 14 q^{7} + 123 q^{8} - 288 q^{9} + 35 q^{10} - 192 q^{11} + 33 q^{12} - 88 q^{13} - 214 q^{14} + 54 q^{15} + 298 q^{16} - 272 q^{17} - 18 q^{18} - 64 q^{19} - 164 q^{20} + 42 q^{21} + 473 q^{22} - 732 q^{23} - 324 q^{24} - 484 q^{25} - 353 q^{26} - 864 q^{27} + 4338 q^{28} + 1646 q^{29} + 666 q^{30} + 302 q^{31} + 910 q^{32} + 612 q^{33} - 138 q^{34} - 552 q^{35} + 99 q^{36} + 3388 q^{37} + 194 q^{38} - 264 q^{39} + 2003 q^{40} + 1152 q^{41} + 18 q^{42} - 2788 q^{43} - 2439 q^{44} - 36 q^{45} - 477 q^{46} - 512 q^{47} + 894 q^{48} + 5020 q^{49} - 4098 q^{50} - 90 q^{51} - 2992 q^{52} - 368 q^{53} - 54 q^{54} - 2492 q^{55} + 2775 q^{56} + 1194 q^{57} + 2088 q^{58} - 364 q^{59} - 8874 q^{60} + 14 q^{61} + 1762 q^{62} - 666 q^{63} - 10293 q^{64} + 5568 q^{65} - 4290 q^{66} - 1562 q^{67} + 4726 q^{68} - 1800 q^{69} - 2498 q^{70} + 6194 q^{71} - 2556 q^{72} - 2154 q^{73} + 3221 q^{74} - 1452 q^{75} - 8038 q^{76} + 1358 q^{77} + 1845 q^{78} + 4924 q^{79} - 2143 q^{80} - 2592 q^{81} + 5080 q^{82} + 2606 q^{83} + 243 q^{84} + 2786 q^{85} + 6437 q^{86} - 144 q^{87} - 2884 q^{88} + 3018 q^{89} + 315 q^{90} + 5506 q^{91} - 15636 q^{92} - 2526 q^{93} + 2358 q^{94} - 140 q^{95} + 1575 q^{96} + 2408 q^{97} - 7221 q^{98} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −4.94279 − 0.471979i 2.52376 + 1.62192i 16.3530 + 3.15178i −1.82165 − 12.6699i −11.7089 − 9.20798i −8.77874 + 12.3280i −41.2286 − 12.1058i 3.73874 + 8.18669i 3.02413 + 63.4843i
4.2 −4.56923 − 0.436309i 2.52376 + 1.62192i 12.8321 + 2.47318i 0.0429643 + 0.298823i −10.8240 − 8.51208i 4.15175 − 5.83032i −22.3209 − 6.55402i 3.73874 + 8.18669i −0.0659346 − 1.38414i
4.3 −4.11427 − 0.392866i 2.52376 + 1.62192i 8.91749 + 1.71870i 2.22379 + 15.4668i −9.74625 − 7.66453i −16.1259 + 22.6457i −4.28917 − 1.25941i 3.73874 + 8.18669i −3.07291 − 64.5082i
4.4 −3.06689 − 0.292852i 2.52376 + 1.62192i 1.46462 + 0.282282i 1.97722 + 13.7518i −7.26511 − 5.71335i 4.23013 − 5.94039i 19.2392 + 5.64914i 3.73874 + 8.18669i −2.03664 − 42.7544i
4.5 −3.05732 − 0.291939i 2.52376 + 1.62192i 1.40655 + 0.271090i −1.09359 − 7.60610i −7.24244 − 5.69552i 20.0509 − 28.1576i 19.3534 + 5.68268i 3.73874 + 8.18669i 1.12295 + 23.5735i
4.6 −1.70394 − 0.162706i 2.52376 + 1.62192i −4.97850 − 0.959527i −0.668012 − 4.64612i −4.03643 − 3.17429i −17.4462 + 24.4998i 21.4658 + 6.30291i 3.73874 + 8.18669i 0.382297 + 8.02539i
4.7 −1.46775 − 0.140153i 2.52376 + 1.62192i −5.72077 − 1.10259i −0.719139 − 5.00172i −3.47694 − 2.73430i −1.20398 + 1.69075i 19.5598 + 5.74327i 3.73874 + 8.18669i 0.354510 + 7.44208i
4.8 −0.0857546 − 0.00818857i 2.52376 + 1.62192i −7.84814 − 1.51261i 2.32615 + 16.1787i −0.203143 − 0.159753i 3.76989 − 5.29407i 1.32187 + 0.388136i 3.73874 + 8.18669i −0.0669974 − 1.40645i
4.9 −0.0473779 − 0.00452404i 2.52376 + 1.62192i −7.85321 − 1.51358i −2.96060 − 20.5914i −0.112233 − 0.0882609i −2.98860 + 4.19691i 0.730545 + 0.214507i 3.73874 + 8.18669i 0.0471106 + 0.988972i
4.10 0.993416 + 0.0948597i 2.52376 + 1.62192i −6.87755 − 1.32554i 0.448481 + 3.11925i 2.35329 + 1.85065i 10.9985 − 15.4453i −14.3666 − 4.21842i 3.73874 + 8.18669i 0.149636 + 3.14126i
4.11 2.23941 + 0.213838i 2.52376 + 1.62192i −2.88618 − 0.556265i 0.118431 + 0.823704i 5.30492 + 4.17183i −8.75517 + 12.2949i −23.6122 − 6.93317i 3.73874 + 8.18669i 0.0890762 + 1.86994i
4.12 2.82303 + 0.269566i 2.52376 + 1.62192i 0.0413796 + 0.00797527i −2.49944 − 17.3840i 6.68743 + 5.25905i −2.70946 + 3.80491i −21.6533 − 6.35798i 3.73874 + 8.18669i −2.36984 − 49.7491i
4.13 2.84022 + 0.271208i 2.52376 + 1.62192i 0.137874 + 0.0265731i 1.10231 + 7.66675i 6.72816 + 5.29108i −17.6977 + 24.8529i −21.5162 − 6.31771i 3.73874 + 8.18669i 1.05152 + 22.0742i
4.14 4.04250 + 0.386012i 2.52376 + 1.62192i 8.33734 + 1.60689i 2.93548 + 20.4167i 9.57621 + 7.53082i 11.8752 − 16.6763i 1.91228 + 0.561495i 3.73874 + 8.18669i 3.98557 + 83.6675i
4.15 4.76735 + 0.455226i 2.52376 + 1.62192i 14.6649 + 2.82643i −1.49068 − 10.3679i 11.2933 + 8.88115i 14.0800 − 19.7726i 31.8658 + 9.35663i 3.73874 + 8.18669i −2.38685 − 50.1061i
4.16 5.06607 + 0.483751i 2.52376 + 1.62192i 17.5756 + 3.38742i 0.408501 + 2.84119i 12.0009 + 9.43764i −9.17020 + 12.8778i 48.3368 + 14.1930i 3.73874 + 8.18669i 0.695068 + 14.5913i
10.1 −3.91689 − 3.73474i −0.426945 − 2.96946i 1.01303 + 21.2662i −4.66810 + 10.2217i −9.41789 + 13.2256i 7.68968 − 31.6973i 47.1027 − 54.3594i −8.63544 + 2.53559i 56.4599 − 22.6031i
10.2 −3.69235 − 3.52065i −0.426945 − 2.96946i 0.857825 + 18.0080i 5.73558 − 12.5592i −8.87801 + 12.4674i −1.77652 + 7.32293i 33.5045 − 38.6663i −8.63544 + 2.53559i −65.3941 + 26.1799i
10.3 −3.14869 − 3.00227i −0.426945 − 2.96946i 0.519968 + 10.9155i −2.82594 + 6.18794i −7.57081 + 10.6317i −4.75860 + 19.6152i 8.34160 − 9.62671i −8.63544 + 2.53559i 27.4759 − 10.9997i
10.4 −2.42146 − 2.30886i −0.426945 − 2.96946i 0.151991 + 3.19068i 5.31778 − 11.6443i −5.82224 + 8.17619i 3.18602 − 13.1329i −10.5294 + 12.1516i −8.63544 + 2.53559i −39.7619 + 15.9183i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.m.a ✓ 320
67.g even 33 1 inner 201.4.m.a ✓ 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.m.a ✓ 320 1.a even 1 1 trivial
201.4.m.a ✓ 320 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{320} + 2 T_{2}^{319} - 95 T_{2}^{318} - 255 T_{2}^{317} + 3404 T_{2}^{316} + \cdots + 74\!\cdots\!64 \) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display